homework20111107

homework20111107 - Give the Laurent series expansion about...

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Math 417, Fall 2011 Exercise Set #7 Turn in: November 7 1. Find the maximum and minimum of | f ( z ) | on the unit disk { z C || z |≤ 1 } , where f ( z )= z 2 2. 2. What is the radius of convergence for the Taylor series of the function f ( z )= 1 z 2 3 z +2 about z = 0? About z =3 i ? 3. This problem has three parts: a) Find the power series representation for e az centered at z =0 ,where a C is any constant b) Show that e z cos z = 1 2 ° e (1+ i ) z + e (1 i ) z ± . c) Find the power series expansion for f ( z )= e z cos z centered at z =0 . 4. By integrating the power series expansion for f ( z )= 1 1+ z 2 term by term, ±nd a power series expansion for arctan z . What is the radius of convergence? 5. By di²erentiating the geometric series for 1 1 z ,±ndthe Taylor series expansion of the following functions around z =0 : a ) 1 (1 z ) 2 b ) 1 (1 z ) 3 c ) 1 (1 z
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Unformatted text preview: Give the Laurent series expansion about z = 1 for the function f ( z ) = z 2 1 + z . 7. Find the Laurent series expansion of f ( z ) = 1 z 3 cos( 1 z 2 ) valid in the region | z | > 0. 8. Find the Frst three terms of the Laurent series for f ( z ) = 1 e z 1 in the region < | z | < 2 using long-division with the Taylor series of e z 1. 9. Find the Laurent series expansion in terms of z of f ( z ) = z 1 + z in the region 1 < | z | < . Write your answer in summation form. 10. Find the Laurent series expansion in terms of z of the function f ( z ) = 1 z ( z 2 + 1) about z = 0 and about z = . 1...
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This note was uploaded on 02/27/2012 for the course MATH 417 taught by Professor Staff during the Fall '11 term at SUNY Albany.

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